This site is supported by donations to The OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A255211 a(n) = n*(n+1)*(7*n+2)/6. 6
 0, 3, 16, 46, 100, 185, 308, 476, 696, 975, 1320, 1738, 2236, 2821, 3500, 4280, 5168, 6171, 7296, 8550, 9940, 11473, 13156, 14996, 17000, 19175, 21528, 24066, 26796, 29725, 32860, 36208, 39776, 43571, 47600, 51870, 56388, 61161, 66196, 71500, 77080, 82943 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This sequence gives the numbers of triangles of all dimensions in polyiamonds of trapezoid configuration. We obtain a trapezoid with 3 equal sizes (n) and the great base has double value (2*n). It is a 3*n^2-iamond. Also sum of n*(n+1)*(2*n+1)/3 quantities of triangles oriented in a direction and n^2*(n+1)/2 of triangles oriented in the opposite direction. It is also half the number of regular hexagons. LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1). FORMULA G.f.: x*(3 + 4*x) / (1 - x)^4. - Colin Barker, Feb 17 2015 a(n) = Sum_{j=0..n-1} (n-j)*(3*n-2*j) = Sum_{j=1..n} j*(n+2*j) for n>0. a(n) = A000292(2*n) - A000292(n). - Bruno Berselli, Sep 22 2016 Sum_{n>=1} 1/a(n) = 21*HarmonicNumber(2/7)/5 - 6/5 = 0.44513027538601361333... . - Vaclav Kotesovec, Sep 22 2016 EXAMPLE From the second comment: a(1)= 2+1, a(2)= 10+6, a(3)= 28+18, a(4)= 60+40. MATHEMATICA Table[n (n + 1) (7 n + 2)/6, {n, 0, 50}] (* Bruno Berselli, Feb 17 2015 *) PROG (PARI) concat(0, Vec(x*(4*x+3)/(x-1)^4 + O(x^100))) \\ Colin Barker, Feb 17 2015 (PARI) vector(50, n, n--; n*(n+1)*(7*n+2)/6) \\ Bruno Berselli, Feb 17 2015 CROSSREFS Partial sums of A022264. Cf. A000292, A000330, A006331, A002411, A033428, A212977. Sequence in context: A092466 A152618 A296947 * A172482 A212564 A222843 Adjacent sequences:  A255208 A255209 A255210 * A255212 A255213 A255214 KEYWORD nonn,easy AUTHOR Luce ETIENNE, Feb 17 2015 EXTENSIONS Edited and extended by Bruno Berselli, Dec 01 2016 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 21 07:08 EDT 2019. Contains 327253 sequences. (Running on oeis4.)